IMG 0871 has three questions IMG 0875 is the example that how i want you to prove limit exist IMG 0876 is the hint or example how to approach last question !

1. Let us prove that xn > 3 for all n.
For n = 1, we have x1 = 4 > 3, Suppose xk > 3 for some k. we see that
xk+1 =
√
3 + 2xk >
√
3 + 3.2 = 3
Hence we proved that xk > 3 implies xk+1 > 3. This shows that xn > 3
for all n.
Therefore xn is bounded below and xn > 3.
Also we see that
x2n − (
√
3 + 2xn)
2
= x2n − 2xn − 3
= (xn − 3)(xn + 1)
> 0, because xn > 3
Hence we see that
x2n > (
√
3 + 2xn)
2
This gives
xn > xn+1
Hence xn is monotonically decreasing. Also xn is bounded...